Simplify the following expression and state the condition under which the simplification is valid. $q = \dfrac{z^2 - 36}{z - 6}$
Solution: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = z$ $ b = \sqrt{36} = -6$ So we can rewrite the expression as: $q = \dfrac{({z} {-6})({z} + {6})} {z - 6} $ We can divide the numerator and denominator by $(z - 6)$ on condition that $z \neq 6$ Therefore $q = z + 6; z \neq 6$